Interpolation techniques are often used in the oil and gas industry to create a surface, sometimes referred to as a horizon, which may be used to locate hydrocarbons in a subsurface formation. Creating a surface from well tops, which are points on a well representing a surface level on a horizon at that point, in a faulted space is always challenging. It is also possible to use secondary information, such as, but not limited to, seismic data, to aid in the creation of the interpolated surface. Ideally, the interpolation should be done in the unfaulted space and then translated back into the faulted domain. This technique, which is commonly referred to as global unfaulting, effectively restores the faulted space to the unfaulted space on a global basis before faulting occurred. However, global unfaulting is very difficult, especially in the presence of many faults.
Refinement gridding is an interpolation technique that translates a coarse approximation of known data into a finer approximation by increasing the number of interpolated nodes on a step-by-step basis. Refinement gridding interpolation therefore, provides an approach to creating a surface in the faulted space directly. It generates very smooth surfaces when there is no fault, and the performance is very good, but it often creates significant undesirable artifacts 102 along the fault and at its edges as illustrated by the interpolated surface 100 in FIG. 1.
Traditional kriging, which is synonymous with optimal prediction, is another interpolation technique, which predicts unknown values from data observed in known locations. Kriging uses variograms to express the spatial variation, and minimizes the error of predicted values, which are estimated by spatial distribution of the predicted values. Traditional Kriging in the presence of a fault can generate cleaner edges along the fault when search neighborhoods are restricted to one side of the fault 202 or the other side of the fault 204 as illustrated by the interpolated surface 200 in FIG. 2. Traditional kriging, however, does not always eliminate artifacts 302 near fault terminations as illustrated by the interpolated surface 300 in FIG. 3.
There is therefore, a need for creating surfaces in a faulted space that reduces the incidental production of artifacts along faults and near fault terminations.